{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Static condensation and hybridization\n",
    "\n",
    "In this notebook, we explore the more advanced capabilities of Firedrake which facilitate the global reduction of finite element systems. The procedure, known as element-wise \"static condensation,\" is well-known within the FEM community. This approach requires the algebraic manipulation of locally assembled matrices and vectors. We will apply this within the context of hybridizing a mixed method.\n",
    "\n",
    "As our running example, we consider the following saddle-point system: find $(\\mathbf{u}, D) \\in V \\times U \\subset H(\\text{div}) \\times L^2$ such that\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\color{#800020}{\\int_\\Omega \\mathbf{w}\\cdot\\mathbf{u}\\,\\mathrm{d}x +\n",
    "    \\beta\\int_\\Omega \\mathbf{w}\\cdot f\\mathbf{u}^\\perp\\,\\mathrm{d}x}\n",
    "    - \\color{#2A52BE}{g\\beta\\int_\\Omega D\\nabla\\cdot\\mathbf{w}\\,\\mathrm{d}x} &= -R\\lbrack \\mathbf{w} \\rbrack,\n",
    "    \\quad \\forall \\mathbf{w} \\in V, \\\\\n",
    "    \\color{#2A52BE}{H\\beta\\int_\\Omega \\phi\\nabla\\cdot\\mathbf{u}\\,\\mathrm{d}x}\n",
    "    + \\color{#CC5500}{\\int_\\Omega \\phi D\\,\\mathrm{d}x} &= -R\\lbrack \\phi \\rbrack,\n",
    "    \\quad \\forall \\phi \\in U,\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where the residual co-vectors are defined as:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    R\\lbrack \\mathbf{w} \\rbrack &= \\int_\\Omega \\mathbf{w}\\cdot\\mathbf{u}^{0}\\,\\mathrm{d}x -\n",
    "    \\beta\\int_\\Omega \\mathbf{w}\\cdot f\\mathbf{u}^{0\\perp}\\,\\mathrm{d}x +\n",
    "    g\\beta\\int_\\Omega D^{0}\\nabla\\cdot\\mathbf{w}\\,\\mathrm{d}x \\\\\n",
    "    R\\lbrack \\phi \\rbrack &= \\int_\\Omega \\phi D^{0}\\,\\mathrm{d}x -\n",
    "    \\beta\\int_\\Omega \\phi\\nabla\\cdot\\mathbf{u}^{0}\\,\\mathrm{d}x.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Here, $\\beta$, $g$, and $H$ are parameters. Within the context of fluid flow on a rotating sphere, $f$ is the Coriolis parameter. $\\mathbf{u}^{0}$ and $D^0$ are previous states of the fields.\n",
    "\n",
    "We require the numerical solution to the saddle-point system:\n",
    "\n",
    "$$\n",
    "    \\mathcal{A}\\mathbf{x} \\equiv\n",
    "    \\begin{bmatrix}\n",
    "        \\color{#800020}{A} & -g\\beta\\color{#2A52BE}{B^T} \\\\\n",
    "        H\\beta\\color{#2A52BE}{B} & \\color{#CC5500}{C}\n",
    "    \\end{bmatrix}\n",
    "    \\begin{Bmatrix}\n",
    "        U \\\\\n",
    "        D\n",
    "    \\end{Bmatrix} =\n",
    "    \\begin{Bmatrix}\n",
    "        -R\\lbrack \\mathbf{w} \\rbrack \\\\\n",
    "        -R\\lbrack \\phi \\rbrack\n",
    "    \\end{Bmatrix} \\equiv \\mathbf{b}\n",
    "$$\n",
    "\n",
    "for the coefficent vectors $U$ and $D$. A challenge for systems of this type is finding a parameter-indepedent solver for which we have good convergence. First, let's set up an example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example implementation\n",
    "\n",
    "For our domain, let's use a spherical mesh with a radius equal to that of the Earth. We begin by importing Firedrake and defining some relevant constants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "import matplotlib.pyplot as plt\n",
    "from firedrake import *\n",
    "\n",
    "R0 = 6371220.0                  # Radius\n",
    "R = Constant(R0)\n",
    "H = Constant(6000.0)            # Mean depth\n",
    "Omega_f = Constant(1.e-4)       # Angular rotation rate\n",
    "g = Constant(10)                # Acceleration due to gravity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's create a mesh with quadrilateral elements. We will want to use Firedrake's `CubedSphereMesh`. As with all builtin meshes, we can find out what arguments we need to pass in:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function CubedSphereMesh in module firedrake.utility_meshes:\n",
      "\n",
      "CubedSphereMesh(radius, refinement_level=0, degree=1, reorder=None, distribution_parameters=None, comm=<mpi4py.MPI.Intracomm object at 0x11fea5870>)\n",
      "    Generate an cubed approximation to the surface of the\n",
      "    sphere.\n",
      "    \n",
      "    :arg radius: The radius of the sphere to approximate.\n",
      "    :kwarg refinement_level: optional number of refinements (0 is a cube).\n",
      "    :kwarg degree: polynomial degree of coordinate space (defaults\n",
      "        to 1: bilinear quads)\n",
      "    :kwarg reorder: (optional), should the mesh be reordered?\n",
      "    :kwarg comm: Optional communicator to build the mesh on (defaults to\n",
      "        COMM_WORLD).\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(CubedSphereMesh)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So now we need to specify the number of refinements (say 4). Let's also set the mesh `degree` to be cubic. With this choice of coordinate space, we can better resolve the actual curvature of the sphere using bendy quadrilateral elements:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "mesh = CubedSphereMesh(radius=R0, refinement_level=4, degree=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now we just initialize the global normals on this mesh:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = SpatialCoordinate(mesh)\n",
    "mesh.init_cell_orientations(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at the mesh:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
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       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
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       "\n",
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       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
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       "    this.root = $('<div/>');\n",
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       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
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       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
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       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
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       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
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       "    this.rubberband = rubberband;\n",
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       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
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       "    this._resize_canvas = function(width, height) {\n",
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       "        canvas_div.css('width', width)\n",
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       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
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       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
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       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
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       "        return fig.toolbar_button_onclick(event['data']);\n",
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       "    function toolbar_mouse_event(event) {\n",
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       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"\" width=\"640\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(111, projection='3d')\n",
    "triplot(mesh, axes=axes);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's define our discrete function spaces. We use a Raviart-Thomas (RT) mixed method on quadrilaterals, taking $V = RTCF_2$ and $U = DQ_1$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<figure>\n",
    "  <center><img src=\"image/rtcf2_dg1.png\" alt=\"rtcf_dg\" style=\"width: 400px;\"/></center>\n",
    "  <center><figcaption>$RTCF_2$ (left) and $DQ_1 (right)$.\n",
    "      Source: <a href=\"http://femtable.org/\">periodic table of finite elements</a>\n",
    "      </figcaption></center>\n",
    "</figure>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "V = FunctionSpace(mesh, \"RTCF\", 2)\n",
    "U = FunctionSpace(mesh, \"DQ\", 1)\n",
    "W = V * U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And create an expression for the Coriolis term:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = 2 * Omega_f * x[2] / R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For our initial profiles for $\\mathbf{u}$ and $D$, we set them in steady rotating state:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\mathbf{u}^0 &= \\frac{u_{\\text{max}}}{R}\\left( -y, x, 0\\right), \\\\\n",
    "    D^0 &= H - \\left(R\\Omega_f u_{\\text{max}} + \\frac{u_{\\text{max}}^2}{2}\\right)\\frac{z^2}{gR^2}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where $u_{\\text{max}} = 40.0 \\text{m}\\text{s}^{-1}$ and $\\Omega_f$ is the planetary rotation rate. In Firedrake, we simply define the expressions in UFL and project/interpolate into the relevant finite element spaces:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_max = Constant(40)\n",
    "u_expr = as_vector([-u_max * x[1] / R, u_max * x[0] / R, 0.0])\n",
    "D_expr = H -  (R * Omega_f * u_max + u_max**2 / 2.0) * (x[2]**2 / (g * R**2))\n",
    "u0 = Function(V).project(u_expr)\n",
    "D0 = Function(U).interpolate(D_expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also need solution functions for $\\mathbf{u}$ and $D$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "wh = Function(W, name=\"w_h\")    # Fields in a mixed function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we set the coefficient parameter $\\beta$ and define some test/trial functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta = Constant(2000.0)\n",
    "u, D = TrialFunctions(W)\n",
    "w, phi = TestFunctions(W)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "UFL has all the right objects to specify our problem. However, for the integral:\n",
    "$$\n",
    "\\begin{equation*}\n",
    "\\int_\\Omega \\mathbf{w}\\cdot f\\mathbf{u}^\\perp\\,\\mathrm{d}x,\n",
    "\\end{equation*}\n",
    "$$\n",
    "we need to be a bit careful. UFL does indeed have a `perp` operator, but it does not work for embedded manifolds. That is okay! Since UFL is all in Python, we can create our own `perp` via a Python function which returns a UFL expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "perp = lambda u: cross(CellNormal(mesh), u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can define our finite element problem in residual form and create a `LinearVariationalProblem`. Here is the problem in residual form:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    0 = F(\\mathbf{u}, D; \\mathbf{w}, \\phi) &=\n",
    "    \\color{#800020}{\\int_\\Omega \\mathbf{w}\\cdot\\mathbf{u}\\,\\mathrm{d}x +\n",
    "    \\beta\\int_\\Omega \\mathbf{w}\\cdot f\\mathbf{u}^\\perp\\,\\mathrm{d}x} \\\\\n",
    "    &- \\color{#2A52BE}{g\\beta\\int_\\Omega D\\nabla\\cdot\\mathbf{w}\\,\\mathrm{d}x} \\\\\n",
    "    &- \\int_\\Omega \\mathbf{w}\\cdot\\mathbf{u}^{0}\\,\\mathrm{d}x\n",
    "    + \\beta\\int_\\Omega \\mathbf{w}\\cdot f\\mathbf{u}^{0\\perp}\\,\\mathrm{d}x \\\\\n",
    "    &- g\\beta\\int_\\Omega D^{0}\\nabla\\cdot\\mathbf{w}\\,\\mathrm{d}x \\\\\n",
    "    &+ \\color{#CC5500}{\\int_\\Omega \\phi D\\,\\mathrm{d}x} +\n",
    "    \\color{#2A52BE}{H\\beta\\int_\\Omega \\phi\\nabla\\cdot\\mathbf{u}\\,\\mathrm{d}x} \\\\\n",
    "    &- \\int_\\Omega \\phi D^{0}\\,\\mathrm{d}x +\n",
    "    \\beta\\int_\\Omega \\phi\\nabla\\cdot\\mathbf{u}^{0}\\,\\mathrm{d}x,\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "and in UFL:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "uD_eqn = (inner(w, u) + beta*inner(w, f*perp(u))\n",
    "          - g*beta*div(w)*D\n",
    "          - inner(w, u0) + beta*inner(w, f*perp(u0))\n",
    "          - g*beta*div(w)*D0\n",
    "          + phi*D + H*beta*phi*div(u)\n",
    "          - phi*D0 + H*beta*phi*div(u0))*dx\n",
    "a = lhs(uD_eqn)\n",
    "L = rhs(uD_eqn)\n",
    "\n",
    "uD_problem = LinearVariationalProblem(a, L, wh, constant_jacobian=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## An approximate Schur-complement preconditioner\n",
    "\n",
    "In exact arithmetic, the inverse of the Schur-complement factorization of $\\mathcal{A}$ is:\n",
    "\n",
    "$$\n",
    "   \\mathcal{A}^{-1} =\n",
    "   \\begin{bmatrix}\n",
    "       I & g\\beta \\color{#800020}{A^{-1}}\\color{#2A52BE}{B^T} \\\\\n",
    "       0 & I\n",
    "   \\end{bmatrix}\n",
    "   \\begin{bmatrix}\n",
    "       \\color{#800020}{A^{-1}} & 0 \\\\\n",
    "       0 & \\color{#CC5500}{S}^{-1}\n",
    "   \\end{bmatrix}\n",
    "   \\begin{bmatrix}\n",
    "       I & 0 \\\\\n",
    "       -H\\beta \\color{#2A52BE}{B}\\color{#800020}{A^{-1}} & I\n",
    "   \\end{bmatrix},\n",
    "$$\n",
    "where $\\color{#CC5500}{S}$ is the Schur-complement:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\color{#CC5500}{S} = \\color{#CC5500}{C} + gH\\beta^2\\color{#2A52BE}{B}\\color{#800020}{A^{-1}}\\color{#2A52BE}{B^T}.\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Problem: $\\color{#800020}{A^{-1}}$ is _dense_! Instead, let's construct a sparse approximation by using the diagonal of $\\color{#800020}{A}$ to form:\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "    \\color{#CC5500}{\\tilde{S}} = \\color{#CC5500}{C} + gH\\beta^2\\color{#2A52BE}{B}\\text{Diag}(\\color{#800020}{A})^{-1}\\color{#2A52BE}{B^T}.\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we use a block-preconditioner (centered around a full factorization) and approximately invert both $\\color{#800020}{A}$ and $\\color{#CC5500}{\\tilde{S}}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "solver_parameters = {'ksp_type': 'gmres',\n",
    "                     'ksp_rtol': 1.0e-7,\n",
    "                     'ksp_max_it': 1500,\n",
    "                     'pc_type': 'fieldsplit',\n",
    "                     'pc_fieldsplit': {'type': 'schur',\n",
    "                                       'schur_fact_type': 'full',\n",
    "                                       'schur_precondition': 'selfp'},\n",
    "                     'fieldsplit_0': {'ksp_type': 'preonly',\n",
    "                                      'pc_type': 'bjacobi',\n",
    "                                      'sub_pc_type': 'ilu'},\n",
    "                     'fieldsplit_1': {'ksp_type': 'preonly',\n",
    "                                      'pc_type': 'gamg',\n",
    "                                      'mg_levels': {'ksp_type': 'chebyshev',\n",
    "                                                    'ksp_max_it': 5,\n",
    "                                                    'pc_type': 'bjacobi',\n",
    "                                                    'sub_pc_type': 'ilu'}}}\n",
    "uD_solver_gmres = LinearVariationalSolver(uD_problem,\n",
    "                                          solver_parameters=solver_parameters)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we just call solve, store our results, and take a look at the reduction in the residual:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "reduction in residual: 3.682334248701547e-10\n"
     ]
    }
   ],
   "source": [
    "wh.assign(0.0)\n",
    "uD_solver_gmres.solve()\n",
    "r = assemble(action(a, wh) - L)\n",
    "b = assemble(L)\n",
    "print(\"reduction in residual: %s\" % (r.dat.norm / b.dat.norm))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solver convergence\n",
    "\n",
    "Although the code we wrote above works fine, let's take a closer look by inspecting solver."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations = 5, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "def gmres_solver_conv(solver):\n",
    "    from firedrake.solving_utils import KSPReasons\n",
    "\n",
    "    print(\"gmres iterations = {}, converged reason = {}\".format(\n",
    "           solver.snes.ksp.getIterationNumber(), \n",
    "           KSPReasons[solver.snes.ksp.getConvergedReason()]))\n",
    "\n",
    "gmres_solver_conv(uD_solver_gmres)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay. But what happens if we change the parameters a bit? For example, increasing $\\beta$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations = 7, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(4000.0)\n",
    "wh.assign(0.0)        # re-initialize solution vector\n",
    "uD_solver_gmres.solve()\n",
    "gmres_solver_conv(uD_solver_gmres)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations = 12, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(8000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_gmres.solve()\n",
    "gmres_solver_conv(uD_solver_gmres)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations = 27, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(16000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_gmres.solve()\n",
    "gmres_solver_conv(uD_solver_gmres)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations = 74, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(32000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_gmres.solve()\n",
    "gmres_solver_conv(uD_solver_gmres)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations = 589, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(64000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_gmres.solve()\n",
    "gmres_solver_conv(uD_solver_gmres)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Yikes. Can we do better?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The hybridized mixed method\n",
    "\n",
    "Hybridizing the mixed formulation is an alternative approach which avoids building a global dense operator. We seek approximations $\\hat{\\mathbf{u}}, D, \\lambda \\in \\widehat{V} \\times U \\times T$, where $\\widehat{V}$ is a _discontinuous_ version of $V$ and $T$ is the _trace_ of $V$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$V = RTCF_2$ is continuous in the sense that nodes on the facets shared between cells are topologically identical. The result of rendering $V$ discontinuous removes this association:\n",
    "\n",
    "<ul class=\"list-unstyled list-inline text-center\">\n",
    "  <li>\n",
    "    <img src='image/w2.png' alt='w2' style=\"width: 275px;\"/>\n",
    "    <figcaption>$V$: continuous between facets.</figcaption>\n",
    "  </li>\n",
    "    &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160;\n",
    "  <li>\n",
    "    <img src='image/w2b.png' alt='w2b' style=\"width: 320px;\"/>\n",
    "    <figcaption>$\\widehat{V}$: no continuity across cell facets.</figcaption>\n",
    "  </li>\n",
    "</ul>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The trace of $V$ is a scalar-valued function space defined only on the mesh skeleton, $\\mathcal{E}$. It is constructed such that function in $T$ belong to the _same_ polynomial space as $\\mathbf{w}\\cdot\\mathbf{n}$, $\\mathbf{w} \\in V$:\n",
    "\n",
    "<figure>\n",
    "    <center><img src='image/w2t.png' alt='w2t' style=\"width: 250px;\"/></center>\n",
    "    <center><figcaption>The space of traces, $T$. Functions in this space are discontinuous across vertices.</figcaption></center>\n",
    "</figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The approximations $\\hat{\\mathbf{u}}, D, \\lambda$ satisfy the following variational problem:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\color{#800020}{\\int_\\Omega \\hat{\\mathbf{w}}\\cdot\\hat{\\mathbf{u}}\\,\\mathrm{d}x +\n",
    "    \\beta\\int_\\Omega \\hat{\\mathbf{w}}\\cdot f\\hat{\\mathbf{u}}^\\perp\\,\\mathrm{d}x}\n",
    "    - \\color{#2A52BE}{g\\beta\\int_\\Omega D\\nabla\\cdot\\hat{\\mathbf{w}}\\,\\mathrm{d}x} +\n",
    "    \\color{#50404D}{\\sum_{K \\in \\Omega} \\int_{\\partial K} \\lambda \\hat{\\mathbf{w}}\\cdot\\mathbf{n}\\,\\mathrm{d}s}\n",
    "    &= -R\\lbrack \\hat{\\mathbf{u}} \\rbrack,\n",
    "    \\quad \\forall \\hat{\\mathbf{w}} \\in \\widehat{V}, \\\\\n",
    "    \\color{#CC5500}{\\int_\\Omega \\phi D\\,\\mathrm{d}x} +\n",
    "    \\color{#2A52BE}{H\\beta\\int_\\Omega \\phi\\nabla\\cdot\\hat{\\mathbf{u}}\\,\\mathrm{d}x} &= -R\\lbrack \\phi \\rbrack,\n",
    "    \\quad \\forall \\phi \\in U, \\\\\n",
    "    \\color{#50404D}{\\sum_{K \\in \\Omega} \\int_{\\partial K} \\gamma \\hat{\\mathbf{u}}\\cdot\\mathbf{n}\\,\\mathrm{d}s}\n",
    "    &= 0, \\quad\\quad\\quad \\forall \\gamma \\in T.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "This is the \"hybridized\" mixed method, resulting in an augmented system with an additional unknown $\\lambda$. This system is solving the same PDE."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The discrete hybridized system\n",
    "\n",
    "The resulting discrete system is:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "    \\color{#800020}{\\hat{A}} & -g\\beta\\color{#2A52BE}{B^T} & \\color{#50404D}{E^T} \\\\\n",
    "    H\\beta\\color{#2A52BE}{B} & \\color{#CC5500}{C} & 0 \\\\\n",
    "    \\color{#50404D}{E} & 0 & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{Bmatrix}\n",
    "    \\hat{U} \\\\\n",
    "    D \\\\\n",
    "    \\Lambda\n",
    "\\end{Bmatrix} =\n",
    "\\begin{Bmatrix}\n",
    "    -R^n\\lbrack \\hat{\\mathbf{w}} \\rbrack \\\\\n",
    "    -R^n\\lbrack \\phi \\rbrack \\\\\n",
    "    0\n",
    "\\end{Bmatrix}.\n",
    "$$\n",
    "\n",
    "Upon initial inspection, it may appear that hybridizing the mixed problem is counter-productive. However, since $\\hat{U}$ and $D$ coupled within the cell interiors only, both can be **eliminated cell-wise** via static condensation!\n",
    "\n",
    "<ul class=\"list-unstyled list-inline text-center\">\n",
    "  <li>\n",
    "    <img src='image/global_mixed_sparsity.png' alt='sparsemat_mixed' style=\"width: 275px;\"/>\n",
    "    <figcaption>Sparsity pattern for the original mixed system.</figcaption>\n",
    "  </li>\n",
    "    &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160;\n",
    "  <li>\n",
    "    <img src='image/global_hybridized_sparsity.png' alt='sparsemat_mixed' style=\"width: 275px;\"/>\n",
    "    <figcaption>Sparsity pattern for the hybridizable system.</figcaption>\n",
    "  </li>\n",
    "</ul>\n",
    "\n",
    "A reduced problem for $\\Lambda$ can be formed:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "    \\color{#50404D}{E} & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    \\color{#800020}{\\hat{A}} & -g\\beta\\color{#2A52BE}{B^T}\\\\\n",
    "    H\\beta\\color{#2A52BE}{B} & \\color{#CC5500}{C}\n",
    "\\end{bmatrix}^{-1}\n",
    "\\begin{bmatrix}\n",
    "    \\color{#50404D}{E^T} \\\\\n",
    "    0\n",
    "\\end{bmatrix}\n",
    "\\Lambda =\n",
    "\\begin{bmatrix}\n",
    "    \\color{#50404D}{E} & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    \\color{#800020}{\\hat{A}} & -g\\beta\\color{#2A52BE}{B^T}\\\\\n",
    "    H\\beta\\color{#2A52BE}{B} & \\color{#CC5500}{C}\n",
    "\\end{bmatrix}^{-1}\n",
    "\\begin{Bmatrix}\n",
    "    -R^n\\lbrack \\hat{\\mathbf{w}} \\rbrack \\\\\n",
    "    -R^n\\lbrack \\phi \\rbrack\n",
    "\\end{Bmatrix}.\n",
    "$$\n",
    "\n",
    "There are a number of advantages to inverting the $\\Lambda$ system over the original mixed problem.\n",
    "* The full hybridized system is **never** explicitly assembled;\n",
    "\n",
    "* The block operator:\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "    \\color{#800020}{\\hat{A}} & -g\\beta\\color{#2A52BE}{B^T}\\\\\n",
    "    H\\beta\\color{#2A52BE}{B} & \\color{#CC5500}{C}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "can be inverted cell-wise.\n",
    "\n",
    "* Once $\\Lambda$ is determined, $\\hat{U}$ and $D$ can be **recovered cell-wise** by inverting the local systems:\n",
    "$$\n",
    "\\begin{Bmatrix}\n",
    "    \\hat{U} \\\\\n",
    "    D\n",
    "\\end{Bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "    \\color{#800020}{\\hat{A}} & -g\\beta\\color{#2A52BE}{B^T}\\\\\n",
    "    H\\beta\\color{#2A52BE}{B} & \\color{#CC5500}{C}\n",
    "\\end{bmatrix}^{-1}\n",
    "\\left(\n",
    "\\begin{Bmatrix}\n",
    "    -R^n\\lbrack \\hat{\\mathbf{w}} \\rbrack \\\\\n",
    "    -R^n\\lbrack \\phi \\rbrack\n",
    "\\end{Bmatrix}\n",
    "-\n",
    "\\begin{bmatrix}\n",
    "    \\color{#50404D}{E^T}\\\\\n",
    "    0\n",
    "\\end{bmatrix}\n",
    "\\Lambda\\right).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "People are often dissuaded from hybridized finite element solvers for their tedious implementation stages. However, Firedrake possesses an abstraction layer for automatically generating element-local dense linear algebra kernels (Slate).\n",
    "In Firedrake, you can write out the dense linear algebra expressions directly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M_1[(2, range(0, 1))]_0 * (PartialPivLU(M_1[(range(0, 1), range(0, 1))]_3)_2).inv * M_1[(range(0, 1), 2)]_4\n"
     ]
    }
   ],
   "source": [
    "Vhat = FunctionSpace(mesh, BrokenElement(V.ufl_element()))\n",
    "Uhat = FunctionSpace(mesh, U.ufl_element())\n",
    "T = FunctionSpace(mesh, FiniteElement(\"HDiv Trace\", mesh.ufl_cell(), 1))\n",
    "Whybrid = Vhat * Uhat * T\n",
    "\n",
    "n = FacetNormal(mesh)\n",
    "uhat, Dhat, lambdar = TrialFunctions(Whybrid)\n",
    "what, phihat, gammar = TestFunctions(Whybrid)\n",
    "\n",
    "# bilinear form (hybrid-mixed system)\n",
    "a_hybrid = ((inner(what, uhat) + beta*inner(what, f*perp(uhat))\n",
    "            - g*beta*div(what)*Dhat\n",
    "            + phihat*Dhat + H*beta*phihat*div(uhat))*dx\n",
    "            + jump(what, n=n)*lambdar('+')*dS\n",
    "            + jump(uhat, n=n)*gammar('+')*dS)\n",
    "\n",
    "# Slate expression for element-wise static condensation\n",
    "AA = Tensor(a_hybrid)\n",
    "A = AA.blocks\n",
    "S = A[2, 0:1] * A[0:1, 0:1].inv * A[0:1, 2]\n",
    "print(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Firedrake's hybridization interface\n",
    "\n",
    "Using Firedrake's symbolic reasoning capability, we can automate the hybridization and static condensation operations using the Python-based preconditioner: `HybridizationPC`. It can be configured through usual PETSc options:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "hybrid_parameters = {'ksp_type': 'preonly',\n",
    "                     'mat_type': 'matfree',\n",
    "                     'pc_type': 'python',\n",
    "                     'pc_python_type': 'firedrake.HybridizationPC',\n",
    "                     # Solver for the trace system\n",
    "                     'hybridization': {'ksp_type': 'gmres',\n",
    "                                       'pc_type': 'gamg',\n",
    "                                       'pc_gamg_sym_graph': True,\n",
    "                                       'ksp_rtol': 1e-7,\n",
    "                                       'mg_levels': {'ksp_type': 'richardson',\n",
    "                                                     'ksp_max_it': 5,\n",
    "                                                     'pc_type': 'bjacobi',\n",
    "                                                     'sub_pc_type': 'ilu'}}}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now one can run the simulation just as before:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "reduction in residual: 8.401132933543049e-09\n"
     ]
    }
   ],
   "source": [
    "beta.assign(2000.0)      # set beta back to initial value\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid = LinearVariationalSolver(uD_problem,\n",
    "                                           solver_parameters=hybrid_parameters)\n",
    "uD_solver_hybrid.solve()\n",
    "r = assemble(action(a, wh) - L)\n",
    "b = assemble(L)\n",
    "print(\"reduction in residual: %s\" % (r.dat.norm / b.dat.norm))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Great! Now how about a closer look at the solver."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solver convergence for the hybridized system\n",
    "\n",
    "Let's stress out the hybridization solver a bit. As before, we write a convenience function for viewing solver convergence:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "def hybrid_solver_conv(solver):\n",
    "    from firedrake.solving_utils import KSPReasons\n",
    "\n",
    "    # get the ksp associated with the trace system\n",
    "    trace_ksp = solver.snes.ksp.getPC().getPythonContext().trace_ksp\n",
    "\n",
    "    print(\"gmres iterations (trace sys) = {}, converged reason = {}\".format(\n",
    "          trace_ksp.getIterationNumber(), \n",
    "          KSPReasons[trace_ksp.getConvergedReason()]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just as we previously did with our approximate Schur-complement preconditioner, let's go ahead and start a sequence $\\beta$-parametrized runs:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations (trace sys) = 4, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(4000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid.solve()\n",
    "hybrid_solver_conv(uD_solver_hybrid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations (trace sys) = 6, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(8000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid.solve()\n",
    "hybrid_solver_conv(uD_solver_hybrid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations (trace sys) = 9, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(16000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid.solve()\n",
    "hybrid_solver_conv(uD_solver_hybrid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations (trace sys) = 18, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(32000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid.solve()\n",
    "hybrid_solver_conv(uD_solver_hybrid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gmres iterations (trace sys) = 41, converged reason = CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "beta.assign(64000.0)\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid.solve()\n",
    "hybrid_solver_conv(uD_solver_hybrid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solver time comparison"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's set up a problem and solve using preconditioned GMRES and with hybridization. Taking a modest value for $\\beta$, we can use builtin notebook magic to time the execution of the solve:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "beta.assign(32000.0);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's time it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "429 ms ± 18 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "wh.assign(0.0)\n",
    "uD_solver_gmres.solve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now for hybridization:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "205 ms ± 4.88 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "wh.assign(0.0)\n",
    "uD_solver_hybrid.solve()"
   ]
  }
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